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Gaussian process approximations for multicolor Pólya urn models

Part of: Physiological, cellular and medical topics Limit theorems

Published online by Cambridge University Press:  25 February 2021

Konstantin Borovkov Open the ORCID record for Konstantin Borovkov [Opens in a new window]
Konstantin Borovkov*
Affiliation:
The University of Melbourne
*
*Postal address: School of Mathematics and Statistics, The University of Melbourne, Parkville3010, Australia. Email address: borovkov@unimelb.edu.au
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Abstract

Motivated by mathematical tissue growth modelling, we consider the problem of approximating the dynamics of multicolor Pólya urn processes that start with large numbers of balls of different colors and run for a long time. Using strong approximation theorems for empirical and quantile processes, we establish Gaussian process approximations for the Pólya urn processes. The approximating processes are sums of a multivariate Brownian motion process and an independent linear drift with a random Gaussian coefficient. The dominating term between the two depends on the ratio of the number of time steps n to the initial number of balls N in the urn. We also establish an upper bound of the form $c(n^{-1/2}+N^{-1/2})$ for the maximum deviation over the class of convex Borel sets of the step-n urn composition distribution from the approximating normal law.

Keywords

multicolor Pólya–Eggenberger urnstrong approximationKiefer processproliferative tissue growthcentral limit theoremconvergence rates

MSC classification

Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 60F15: Strong theorems 60F05: Central limit and other weak theorems 92C17: Cell movement (chemotaxis, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 58 , Issue 1 , March 2021 , pp. 274 - 286
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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